Properties

Label 1305.2.a.m
Level $1305$
Weight $2$
Character orbit 1305.a
Self dual yes
Analytic conductor $10.420$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + (\beta + 3) q^{4} - q^{5} + q^{7} + (2 \beta + 5) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (\beta + 3) q^{4} - q^{5} + q^{7} + (2 \beta + 5) q^{8} - \beta q^{10} - 5 q^{11} + (2 \beta - 1) q^{13} + \beta q^{14} + (5 \beta + 4) q^{16} + 3 q^{17} - 2 \beta q^{19} + ( - \beta - 3) q^{20} - 5 \beta q^{22} + 4 q^{23} + q^{25} + (\beta + 10) q^{26} + (\beta + 3) q^{28} - q^{29} + 4 q^{31} + (5 \beta + 15) q^{32} + 3 \beta q^{34} - q^{35} - 4 q^{37} + ( - 2 \beta - 10) q^{38} + ( - 2 \beta - 5) q^{40} + ( - 4 \beta + 2) q^{41} + (2 \beta - 6) q^{43} + ( - 5 \beta - 15) q^{44} + 4 \beta q^{46} + (2 \beta - 7) q^{47} - 6 q^{49} + \beta q^{50} + (7 \beta + 7) q^{52} + ( - 2 \beta - 4) q^{53} + 5 q^{55} + (2 \beta + 5) q^{56} - \beta q^{58} + ( - 2 \beta + 4) q^{59} + ( - 6 \beta + 2) q^{61} + 4 \beta q^{62} + (10 \beta + 17) q^{64} + ( - 2 \beta + 1) q^{65} + (4 \beta + 3) q^{67} + (3 \beta + 9) q^{68} - \beta q^{70} + ( - 2 \beta + 6) q^{71} + 4 q^{73} - 4 \beta q^{74} + ( - 8 \beta - 10) q^{76} - 5 q^{77} + ( - 2 \beta + 4) q^{79} + ( - 5 \beta - 4) q^{80} + ( - 2 \beta - 20) q^{82} + ( - 2 \beta + 8) q^{83} - 3 q^{85} + ( - 4 \beta + 10) q^{86} + ( - 10 \beta - 25) q^{88} + ( - 2 \beta - 5) q^{89} + (2 \beta - 1) q^{91} + (4 \beta + 12) q^{92} + ( - 5 \beta + 10) q^{94} + 2 \beta q^{95} + ( - 2 \beta + 8) q^{97} - 6 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 7 q^{4} - 2 q^{5} + 2 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 7 q^{4} - 2 q^{5} + 2 q^{7} + 12 q^{8} - q^{10} - 10 q^{11} + q^{14} + 13 q^{16} + 6 q^{17} - 2 q^{19} - 7 q^{20} - 5 q^{22} + 8 q^{23} + 2 q^{25} + 21 q^{26} + 7 q^{28} - 2 q^{29} + 8 q^{31} + 35 q^{32} + 3 q^{34} - 2 q^{35} - 8 q^{37} - 22 q^{38} - 12 q^{40} - 10 q^{43} - 35 q^{44} + 4 q^{46} - 12 q^{47} - 12 q^{49} + q^{50} + 21 q^{52} - 10 q^{53} + 10 q^{55} + 12 q^{56} - q^{58} + 6 q^{59} - 2 q^{61} + 4 q^{62} + 44 q^{64} + 10 q^{67} + 21 q^{68} - q^{70} + 10 q^{71} + 8 q^{73} - 4 q^{74} - 28 q^{76} - 10 q^{77} + 6 q^{79} - 13 q^{80} - 42 q^{82} + 14 q^{83} - 6 q^{85} + 16 q^{86} - 60 q^{88} - 12 q^{89} + 28 q^{92} + 15 q^{94} + 2 q^{95} + 14 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.79129
2.79129
−1.79129 0 1.20871 −1.00000 0 1.00000 1.41742 0 1.79129
1.2 2.79129 0 5.79129 −1.00000 0 1.00000 10.5826 0 −2.79129
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(29\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1305.2.a.m 2
3.b odd 2 1 435.2.a.f 2
5.b even 2 1 6525.2.a.t 2
12.b even 2 1 6960.2.a.bw 2
15.d odd 2 1 2175.2.a.r 2
15.e even 4 2 2175.2.c.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.f 2 3.b odd 2 1
1305.2.a.m 2 1.a even 1 1 trivial
2175.2.a.r 2 15.d odd 2 1
2175.2.c.f 4 15.e even 4 2
6525.2.a.t 2 5.b even 2 1
6960.2.a.bw 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1305))\):

\( T_{2}^{2} - T_{2} - 5 \) Copy content Toggle raw display
\( T_{7} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 5 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( (T + 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 21 \) Copy content Toggle raw display
$17$ \( (T - 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 20 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( (T + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 84 \) Copy content Toggle raw display
$43$ \( T^{2} + 10T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 15 \) Copy content Toggle raw display
$53$ \( T^{2} + 10T + 4 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T - 12 \) Copy content Toggle raw display
$61$ \( T^{2} + 2T - 188 \) Copy content Toggle raw display
$67$ \( T^{2} - 10T - 59 \) Copy content Toggle raw display
$71$ \( T^{2} - 10T + 4 \) Copy content Toggle raw display
$73$ \( (T - 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 6T - 12 \) Copy content Toggle raw display
$83$ \( T^{2} - 14T + 28 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 15 \) Copy content Toggle raw display
$97$ \( T^{2} - 14T + 28 \) Copy content Toggle raw display
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